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ENEE 241 02
*
READING ASSIGNMENT 15
Tue 04/07 Lecture
Topics:
signal transformations and the DFT; duality; circular symmetry revisited
Textbook References:
sections 3.4, 3.5
Key Points:
Time Domain
Frequency Domain
x
=
1
N
VX
←→
X
=
Wx
P
m
x
←→
F

m
X
F
m
x
←→
P
m
X
X
←→
N
Rx
•
Every DFT pair has a dual: if
x
←→
X
, then
X
←→
N
Rx
.
•
Circular conjugate symmetry in either domain (time or frequency) implies exclusively real
values in the other domain, and vice versa. If a signal is both realvalued and circularly
symmetric, then so is its spectrum.
Theory and Examples:
1
. Let us take a closer look at circular shifts of the rows and/or columns of
V
and
W
. The
diagonal matrix
F
(whose leading diagonal consists of the entries of
v
(1)
) is implicitly involved
here. If
1
is the allones column vector, then we can write
V
=
£
1 F1 F
2
1
...
F
N

1
1
/
A circular shift on the columns of
V
is obtained by rightmultiplying it by
P
T
=
P

1
,
resulting in
VP

1
=
£
F
N

1
1 1 F
1
1
...
F
N

2
1
/
=
£
F

1
1 1 F
1
1
...
F
N

2
1
/
=
F

1
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 Spring '08
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